STEP-BY-STEP LESSON · §3.4
Arguments with Quantified Statements
Apply a general rule to a specific admissible object.
Before you begin
Valid and Invalid Arguments
An argument is valid when its premises cannot all be true while its conclusion is false.
From p→q and p infer q: the rule’s hypothesis is satisfied. This is modus ponens.
p→q, p ∴ q
Statements with Multiple Quantifiers
In ∀x∃y, x is given first, then a suitable y may be selected. y may change with x.
∀x∈ℤ ∃y∈ℤ: x+y=0
In ∃y∀x, fix one y first; it must work for every x. You cannot choose a new y after each x.
∃y∈ℤ ∀x∈ℤ: x+y=0
Symbols
- ∀
- for every
- ∴
- therefore
Step by step
Step 1 / 6
Instantiate a general rule
From ∀x∈D P(x), infer P(a) for a selected a∈D. Domain membership is part of the justification.
∀x∈D P(x), a∈D ∴ P(a)
The rule applies only within its domain.
Worked example
Every object in D with P has Q. Some object in D has P. Show that some object in D has Q.
- Choose a witness a∈D with P(a), justified by the existential premise.
- Instantiate the universal rule at a: P(a)→Q(a). This is allowed because a∈D.
- Use P(a) to conclude Q(a), then use a as a witness for ∃x∈D Q(x).
Optional self-check
Can P(2) imply ∀n∈ℤ P(n)?
Show answer
No, a special case does not establish a universal claim.